I would agree that Np => p does not hold in all modal logics, such as
denotic or provability logics. However, it does appears to be the
defining minimal characteristic for logics of necessity. It is the
basis of the classical system T. There are many other modal logics in
which this doesn't hold, but they aren't generally regarded as logics
of necessity.
It appears to me that you confuse the general idea of modal logics
with the more specific idea of necessity logics. So if Np => p is not
validated in all necessity logics, what properties do you see as
defining the minimal properties of necessity -- as opposed to the
many other modal logics? You could have Np => Pp as a weaker
condition (the basis of the classical system D), but that is usually
considered to be basis of denotic logic (at least when Np => p is not
also present).
Michael Lee Finney
RH> On 07/04/2010 11:40 AM, Michael Lee Finney wrote:
>> It would seem to me that Np means that p is necessarily true without
>> distinction of world, or that p is true in all possible worlds. In
>> either case it would be true in the actual world because in one case
>> the world was not distinguished and in the other case surely the
>> actual world is possible. So you have Np => Np[a] no matter how you
>> look at it.
>>>>RH> In general, this is not true; that is, it is not true when we think of
RH> necessity in very general terms.
RH> In (propositional) modal logic, a model is a quadruple <W,R,V,@> where W
RH> is a set of worlds, R is a relation of "accessibility" between the
RH> worlds, V is a valuation function, which we can think of as a map from
RH> worlds to sentence letters true in those worlds, and @ (a member of W)
RH> is the actual world. A formula "N\phi", commonly written "\Box\phi", is
RH> true at a world W in the model if \phi is true at all worlds that are
RH> _accessible from_ w. One can therefore have "N\phi" be true at a world
RH> without having \phi be true at all worlds; since this can happen at @,
RH> "N\phi" can be true in the model without its being true at all worlds in
RH> the model.
RH> This is important, as it is not obvious that every reasonable notion of
RH> necessity must validate, for example, "N\phi --> NN\phi". Why shouldn't
RH> there be a proposition that is, as things stand, necessary but that, had
RH> things been otherwise, might have been contingent? Why shouldn't there
RH> be a proposition that isn't, as things stand, necessary but, had things
RH> been otherwise, might have been necessary? Anyone who thinks that there
RH> are contingent non-identities but who thinks (with Kripke) that all
RH> identities are necessary holds this latter view.
RH> Moreover, since the accessibility relation R need not be reflexive,
RH> "N\phi" could be true at a world without \phi being true at that world.
RH> Again, this can happen at @, so "N\phi" need not imply \phi, in general.
RH> Of course, one might sensibly argue that any reasonable notion of
RH> necessity must validate "N\phi --> \phi" and so argue that any modal
RH> logic modelling any reasonable notion of necessity must have only models
RH> in which the accessibility relation is reflexive. But that is a
RH> substantive---i.e., not purely logical---claim.
RH> Richard Heck
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